Day 15

Math 216: Statistical Thinking

Bastola

Exponential Distribution

• Models waiting times between events (time/distance)
  

• Key applications:

  ▸ Emergency intervals
  ▸ Equipment failures
  ▸ Rare species sightings

• Defining features:

  • Single parameter \(θ\) controls both shape and scale
    
  • Memoryless property (future probability independent of past)

• Alternate name: “Waiting-time distribution”

Probability Density Function

\[f(x)=\frac{1}{\theta} \exp \left(-\frac{x}{\theta}\right), \quad x>0\]

• \(\theta\): Scale parameter, affects the decay rate of the function

Parameters of the Exponential Distribution

• Mean: \(\mu=\theta\)
• Rate: \(\lambda=\frac{1}{\theta}\)
• Standard Deviation: \(\sigma=\theta\)
• Variance: \(\sigma^2=\theta^2\)

Visualizing the Exponential Distribution

• The shape becomes wider as \(\theta\) increases
• Example: Plots for \(\theta=0.5, 1, 2\)

Calculating Probabilities

• To find the area to the right of a value \(a\): \(P(X > a) = \exp\left(-\frac{a}{\theta}\right)\)

Calculating Probabilities and Quantiles in R

pexp: Calculates the cumulative probability for a given value.

• Syntax: pexp(q, rate = 1, lower.tail = TRUE)
• Computes \(P(X \leq x)\) or \(P(X > x)\) for an exponential distribution.
• Examples:
  • pexp(5, rate = 1/2) calculates \(P(X \leq 5)\) for \(\theta = 2\).
  • pexp(5, rate = 1/2, lower.tail = FALSE) for \(P(X > 5)\).

Calculating Probabilities and Quantiles in R

qexp: Computes the quantile for a specified probability.

• Syntax: qexp(p, rate = 1, lower.tail = TRUE)
• Finds the smallest \(x\) such that \(P(X \leq x) \geq p\).
• Examples:
  • qexp(0.7, rate = 1/2) finds the time by which 70% of events are expected to occur for \(\theta = 2\).

Application Examples

• Hospital Emergencies:
  • Given \(\theta = 2\), find \(P(X > 5)\):

1 - pexp(5, rate = 1/2)  # Method 1

[1] 0.082085

pexp(5, rate = 1/2, lower.tail = FALSE)  # Method 2

[1] 0.082085

• Warranty Period for Magnetron Tubes:
  • With \(\theta = 6.25\), find the quantile for 90% reliability:

qexp(0.9, rate = 1/6.25)

[1] 14.39116